Topics

Introduction to :

No offense to prisms and cylinders. They're great. But we don't eat our ice cream out of prisms while touring the Egyptian Cylinders of Giza. What we're saying is that there's no substitute for pyramids and cones, so we might as well learn a little bit about them.

The pyramid is a regular pyramid if the base of the pyramid is a regular polygon with a vertex that is perpendicular to the center of the base. A pyramid is irregular if it doesn't have daily bowel movements.

The height of the overall pyramid is called the altitude but the lateral faces have their own height called the slant height. Let's take a regular hexagonal pyramid and look at its net (which looks surprisingly like the Star of David. L'chaim!).

The lateral area is just the area of all the triangles combined. The area of a triangle is one half times the base of the triangle times the height. In this case, the base is the side of the hexagon s and the height of the triangle is the slant height l. Since we have six of these triangles, the lateral area is given by this equation:

Or, simplified, it equals:

Since we know the perimeter of the hexagon equals 6s, we can replace the sides with the perimeter of the hexagon P.

Mazel tov! We've found the lateral area formula for any pyramid.

All that's missing for us to find the surface area is the base of the pyramid, B. Remember that unlike cylinders and prisms, pyramids only have one base. If we know the area of the base, we can find out the surface area of the whole pyramid.

Sample Problem

For Father's Day, you decide to build your dad a birdhouse. You know how much he loves birds and you'd rather do that than be his golf caddy for the millionth time. (Seriously, how many times can he all you "champ" in one day?) You decide that roof of the birdhouse will be a square pyramid because it's the easiest. The slant height of the roof is 10 inches and each side of the square is 8 inches. How much wood in square inches will you need to make the roof?

Well, we can start with our lateral area formula.

The perimeter of the square base is 8 + 8 + 8 + 8 = 32 inches. The slant height that you want for the roof is 10 inches.

Just be careful with that hacksaw. You don't want to end up like that woodshop teacher with the missing pinky.

We can use far more than these formulas to figure out surface area. Remember the Pythagorean theorem? Remember trigonometry? Those mathematical ghosts will never leave you. As long as we treat them right, they'll be more like Casper and less like the Exorcist.

Sample Problem

What is the surface area of this pentagonal pyramid?

Knowing the altitude and the slant height can give us enough information to calculate everything we need. If we look at those two values as a right triangle, the altitude is a leg and the slant height is the hypotenuse. Solving for the remaining leg (and hopefully it isn't prosthetic) will give us the distance from the center of the base to one of the sides.

a2 + b2 = c2a2 + 122 = 132a2 = 169 – 144a2 = 25a = 5

Zoom in on the base. A perpendicular line segment from the center to the side is 5 inches. Since the pentagon is regular, we can also find the angles in the right triangle. The 360° around the center are split into 5 evenly, which give us 72° per isosceles triangle. We're splitting these triangles in half, so each angle is 36°. Here's what we know, Pictionary style.

Let's not go off on a tangent here. Well...actually…

We can solve for s and find the length of each side of the pentagon.

s = 10tan(36°) ≈ 7.27 inches

Be sure to thank trigonometry in your Nobel Prize acceptance speech.

Now we can do two things:

Find the area of the pentagonal base

Find the perimeter, which will help us find the lateral area

Area is first, so we'll start with that.

If you imagine cutting the pentagon into 5 identical triangles (feel free to imagine cutting it a little more if you're really mad at it), we can find the area of each triangle and multiply it by 5. The base of the triangle is the side length and the height is 5.

Halfway done. If this were a romantic comedy, we'd be at the point where the two main characters have fallen in love already but don't know it yet, and some random conflict has just happened that tears them apart. That leaves just enough time for the epiphany of their love, the frantic last-minute chase for each other, and the obligatory saliva swap at the end.

The base is a pentagon ("penta-" means five), so perimeter is five times the length of the side. We know the side is about 7.27 inches, which means the perimeter is about 36.3 inches. High penta!

Now for the part where the two main characters realize their undying love and race to find each other. The lateral area of a pyramid is given by the formula:

We know the slant height l since it was given to us, and we just calculated P.

The two lovers have reunited and professed their love at long last. All that remains is that slobbery kiss that lasts for centuries.

Example 1

To find the lateral surface area, we need to know the perimeter of the base and the slant height. The perimeter is 0.8 × 6 = 4.8, but we need to futz around a bit to find the slant height.

If we split the hexagonal base into six triangles and then bisect them, we'll end up with twelve identical right triangles. The angles facing the center of the hexagon are . That means we have a 30°-60°-90° triangle with the shortest side equaling 0.4 inches (half of the hexagon's side), the hypotenuse equaling 0.8 inches (twice that), and the long side equaling . This might seem pointless, but just go with us.

If we look right at the pyramid instead of down onto it, we see that the altitude and the side (perpendicular to the hexagon edge) create a right triangle with the slant height as the hypotenuse. We can smell the Pythagorean theorem from a mile away.

a2 + b2 = c2

Substitute in our values. The two legs are the altitude (3 inches) and the distance from the center to the side ( inches). The slant height is the hypotenuse.

l2 = 9 + (0.16 × 3)l ≈ 3.08 in2

With the slant height, now we can find the lateral area. We use ½ in the lateral area formula for pyramids because pyramids halve triangles.

That's lateral area. Surface area is just the lateral area plus the area of the base. Since it's a hexagon, the base is going to be a bit tricky. We might not know hexagons that well (nothing against them), but triangles have been with us since the dawn of time. Six triangles have ½ base times height? We know the deal.

Example 2

Find the surface area of the entire figure.

This figure combines our knowledge of the surface area of pyramids and prisms. Oooh, fancy.

First, let's figure out exactly what we want. The surface area of the overall figure consists of the lateral area of the pyramid and the lateral area of the prism along with one base (the bottom). We'll say the formula looks like this:

SA = LPyramid + LPrism + B

Okay…but what do these things even mean? We'll start with the lateral area of the prism. Like we said in the last chapter, it's a folded rectangle where the length is the perimeter of the base and the height is...the height. The perimeter is 15 + 15 + 15 + 15 = 60 ft and the height is 20 ft.

LPrism = PhLPrism = (60 ft)(20 ft) = 1200 ft2

Moving on to the lateral area of the pyramid. Pyramids halve triangles, remember?

The perimeter is the same, but l is the slant height, which is 15 ft.

Two thirds done already? That's awesome.

Trivia Time: Do we have to find the base of the prism or of the pyramid? It's actually a trick question because they're both the same. The base is a square no matter how you look at it.

Exercise 3

The four identical pyramids, the faces of the prism, and the base are all included in the surface area.

Answer

SA ≈ 3425 m2

Exercise 4

Rabbi Schwartz noticed that the net of a regular hexagonal pyramid made a shape similar to the Star of David, so he decided to build a synagogue in that shape. If he wants to paint the building light blue, how much area will he cover?